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_root_.measure_theory.integrable.comp_snd_map_prod_id [normed_add_comm_group F]
(hm : m ≤ mΩ) (hf : integrable f μ) :
integrable (λ x : Ω × Ω, f x.2)
(@measure.map Ω (Ω × Ω) (m.prod mΩ) mΩ (λ ω, (id ω, id ω)) μ) | begin
rw ← integrable_comp_snd_map_prod_mk_iff (measurable_id'' hm) at hf,
simp_rw [id.def] at hf ⊢,
exact hf,
end | lemma | measure_theory.integrable.comp_snd_map_prod_id | probability.kernel | src/probability/kernel/condexp.lean | [
"probability.kernel.cond_distrib"
] | [
"normed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
condexp_kernel (μ : measure Ω) [is_finite_measure μ] (m : measurable_space Ω) :
@kernel Ω Ω m mΩ | @cond_distrib Ω Ω Ω _ mΩ _ _ _ mΩ m id id μ _ | def | probability_theory.condexp_kernel | probability.kernel | src/probability/kernel/condexp.lean | [
"probability.kernel.cond_distrib"
] | [
"measurable_space"
] | Kernel associated with the conditional expectation with respect to a σ-algebra. It satisfies
`μ[f | m] =ᵐ[μ] λ ω, ∫ y, f y ∂(condexp_kernel μ m ω)`.
It is defined as the conditional distribution of the identity given the identity, where the second
identity is understood as a map from `Ω` with the σ-algebra `mΩ` to `Ω` ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
measurable_condexp_kernel {s : set Ω} (hs : measurable_set s) :
measurable[m] (λ ω, condexp_kernel μ m ω s) | by { rw condexp_kernel, convert measurable_cond_distrib hs, rw measurable_space.comap_id, } | lemma | probability_theory.measurable_condexp_kernel | probability.kernel | src/probability/kernel/condexp.lean | [
"probability.kernel.cond_distrib"
] | [
"measurable",
"measurable_set",
"measurable_space.comap_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.measure_theory.ae_strongly_measurable.integral_condexp_kernel
[normed_space ℝ F] [complete_space F]
(hm : m ≤ mΩ) (hf : ae_strongly_measurable f μ) :
ae_strongly_measurable (λ ω, ∫ y, f y ∂(condexp_kernel μ m ω)) μ | begin
rw condexp_kernel,
exact ae_strongly_measurable.integral_cond_distrib
(ae_measurable_id'' μ hm) ae_measurable_id (hf.comp_snd_map_prod_id hm),
end | lemma | measure_theory.ae_strongly_measurable.integral_condexp_kernel | probability.kernel | src/probability/kernel/condexp.lean | [
"probability.kernel.cond_distrib"
] | [
"ae_measurable_id",
"complete_space",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ae_strongly_measurable'_integral_condexp_kernel [normed_space ℝ F] [complete_space F]
(hm : m ≤ mΩ) (hf : ae_strongly_measurable f μ) :
ae_strongly_measurable' m (λ ω, ∫ y, f y ∂(condexp_kernel μ m ω)) μ | begin
rw condexp_kernel,
have h := ae_strongly_measurable'_integral_cond_distrib
(ae_measurable_id'' μ hm) ae_measurable_id (hf.comp_snd_map_prod_id hm),
rwa measurable_space.comap_id at h,
end | lemma | probability_theory.ae_strongly_measurable'_integral_condexp_kernel | probability.kernel | src/probability/kernel/condexp.lean | [
"probability.kernel.cond_distrib"
] | [
"ae_measurable_id",
"complete_space",
"measurable_space.comap_id",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.measure_theory.integrable.condexp_kernel_ae
(hm : m ≤ mΩ) (hf_int : integrable f μ) :
∀ᵐ ω ∂μ, integrable f (condexp_kernel μ m ω) | begin
rw condexp_kernel,
exact integrable.cond_distrib_ae (ae_measurable_id'' μ hm)
ae_measurable_id (hf_int.comp_snd_map_prod_id hm),
end | lemma | measure_theory.integrable.condexp_kernel_ae | probability.kernel | src/probability/kernel/condexp.lean | [
"probability.kernel.cond_distrib"
] | [
"ae_measurable_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.measure_theory.integrable.integral_norm_condexp_kernel
(hm : m ≤ mΩ) (hf_int : integrable f μ) :
integrable (λ ω, ∫ y, ‖f y‖ ∂(condexp_kernel μ m ω)) μ | begin
rw condexp_kernel,
exact integrable.integral_norm_cond_distrib (ae_measurable_id'' μ hm)
ae_measurable_id (hf_int.comp_snd_map_prod_id hm),
end | lemma | measure_theory.integrable.integral_norm_condexp_kernel | probability.kernel | src/probability/kernel/condexp.lean | [
"probability.kernel.cond_distrib"
] | [
"ae_measurable_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.measure_theory.integrable.norm_integral_condexp_kernel
[normed_space ℝ F] [complete_space F]
(hm : m ≤ mΩ) (hf_int : integrable f μ) :
integrable (λ ω, ‖∫ y, f y ∂(condexp_kernel μ m ω)‖) μ | begin
rw condexp_kernel,
exact integrable.norm_integral_cond_distrib (ae_measurable_id'' μ hm)
ae_measurable_id (hf_int.comp_snd_map_prod_id hm),
end | lemma | measure_theory.integrable.norm_integral_condexp_kernel | probability.kernel | src/probability/kernel/condexp.lean | [
"probability.kernel.cond_distrib"
] | [
"ae_measurable_id",
"complete_space",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.measure_theory.integrable.integral_condexp_kernel [normed_space ℝ F] [complete_space F]
(hm : m ≤ mΩ) (hf_int : integrable f μ) :
integrable (λ ω, ∫ y, f y ∂(condexp_kernel μ m ω)) μ | begin
rw condexp_kernel,
exact integrable.integral_cond_distrib (ae_measurable_id'' μ hm)
ae_measurable_id (hf_int.comp_snd_map_prod_id hm),
end | lemma | measure_theory.integrable.integral_condexp_kernel | probability.kernel | src/probability/kernel/condexp.lean | [
"probability.kernel.cond_distrib"
] | [
"ae_measurable_id",
"complete_space",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_to_real_condexp_kernel (hm : m ≤ mΩ) {s : set Ω} (hs : measurable_set s) :
integrable (λ ω, (condexp_kernel μ m ω s).to_real) μ | begin
rw condexp_kernel,
exact integrable_to_real_cond_distrib (ae_measurable_id'' μ hm) hs,
end | lemma | probability_theory.integrable_to_real_condexp_kernel | probability.kernel | src/probability/kernel/condexp.lean | [
"probability.kernel.cond_distrib"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
condexp_ae_eq_integral_condexp_kernel [normed_space ℝ F] [complete_space F]
(hm : m ≤ mΩ) (hf_int : integrable f μ) :
μ[f | m] =ᵐ[μ] λ ω, ∫ y, f y ∂(condexp_kernel μ m ω) | begin
have hX : @measurable Ω Ω mΩ m id := measurable_id.mono le_rfl hm,
rw condexp_kernel,
refine eventually_eq.trans _ (condexp_ae_eq_integral_cond_distrib_id hX hf_int),
simp only [measurable_space.comap_id, id.def],
end | lemma | probability_theory.condexp_ae_eq_integral_condexp_kernel | probability.kernel | src/probability/kernel/condexp.lean | [
"probability.kernel.cond_distrib"
] | [
"complete_space",
"le_rfl",
"measurable",
"measurable_space.comap_id",
"normed_space"
] | The conditional expectation of `f` with respect to a σ-algebra `m` is almost everywhere equal to
the integral `∫ y, f y ∂(condexp_kernel μ m ω)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sequence_anti : antitone (f ∘ (hf.sequence f)) | antitone_nat_of_succ_le $ hf.sequence_mono_nat | lemma | directed.sequence_anti | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"antitone",
"antitone_nat_of_succ_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sequence_le (a : α) : f (hf.sequence f (encodable.encode a + 1)) ≤ f a | hf.rel_sequence a | lemma | directed.sequence_le | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_Inter {s : set α} {t : ι → set β} [hι : nonempty ι] :
s ×ˢ (⋂ i, t i) = ⋂ i, s ×ˢ (t i) | begin
ext x,
simp only [mem_prod, mem_Inter],
exact ⟨λ h i, ⟨h.1, h.2 i⟩, λ h, ⟨(h hι.some).1, λ i, (h i).2⟩⟩,
end | lemma | prod_Inter | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.Union_Iic_rat : (⋃ r : ℚ, Iic (r : ℝ)) = univ | begin
ext1,
simp only [mem_Union, mem_Iic, mem_univ, iff_true],
obtain ⟨r, hr⟩ := exists_rat_gt x,
exact ⟨r, hr.le⟩,
end | lemma | real.Union_Iic_rat | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"exists_rat_gt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.Inter_Iic_rat : (⋂ r : ℚ, Iic (r : ℝ)) = ∅ | begin
ext1,
simp only [mem_Inter, mem_Iic, mem_empty_iff_false, iff_false, not_forall, not_le],
exact exists_rat_lt x,
end | lemma | real.Inter_Iic_rat | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"exists_rat_lt",
"not_forall"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
at_bot_le_nhds_bot {α : Type*} [topological_space α] [linear_order α] [order_bot α]
[order_topology α] :
(at_bot : filter α) ≤ 𝓝 ⊥ | begin
casesI subsingleton_or_nontrivial α,
{ simp only [nhds_discrete, le_pure_iff, mem_at_bot_sets, mem_singleton_iff,
eq_iff_true_of_subsingleton, implies_true_iff, exists_const], },
have h : at_bot.has_basis (λ _ : α, true) Iic := @at_bot_basis α _ _,
have h_nhds : (𝓝 ⊥).has_basis (λ a : α, ⊥ < a) (λ ... | lemma | at_bot_le_nhds_bot | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"eq_iff_true_of_subsingleton",
"exists_const",
"filter",
"nhds_bot_basis",
"nhds_discrete",
"order_bot",
"order_topology",
"subset_trans",
"subsingleton_or_nontrivial",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
at_top_le_nhds_top {α : Type*} [topological_space α] [linear_order α] [order_top α]
[order_topology α] :
(at_top : filter α) ≤ 𝓝 ⊤ | @at_bot_le_nhds_bot αᵒᵈ _ _ _ _ | lemma | at_top_le_nhds_top | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"at_bot_le_nhds_bot",
"filter",
"order_top",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_of_antitone {ι α : Type*} [preorder ι] [topological_space α]
[conditionally_complete_linear_order α] [order_topology α] {f : ι → α} (h_mono : antitone f) :
tendsto f at_top at_bot ∨ (∃ l, tendsto f at_top (𝓝 l)) | @tendsto_of_monotone ι αᵒᵈ _ _ _ _ _ h_mono | lemma | tendsto_of_antitone | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"antitone",
"conditionally_complete_linear_order",
"order_topology",
"tendsto_of_monotone",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ennreal.of_real_cinfi (f : α → ℝ) [nonempty α] :
ennreal.of_real (⨅ i, f i) = ⨅ i, ennreal.of_real (f i) | begin
by_cases hf : bdd_below (range f),
{ exact monotone.map_cinfi_of_continuous_at ennreal.continuous_of_real.continuous_at
(λ i j hij, ennreal.of_real_le_of_real hij) hf, },
{ symmetry,
rw [real.infi_of_not_bdd_below hf, ennreal.of_real_zero, ← ennreal.bot_eq_zero, infi_eq_bot],
obtain ⟨y, hy_mem... | lemma | ennreal.of_real_cinfi | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"bdd_below",
"ennreal.bot_eq_zero",
"ennreal.of_real",
"ennreal.of_real_le_of_real",
"ennreal.of_real_zero",
"infi_eq_bot",
"monotone.map_cinfi_of_continuous_at",
"real.infi_of_not_bdd_below"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lintegral_infi_directed_of_measurable {mα : measurable_space α} [countable β]
{f : β → α → ℝ≥0∞} {μ : measure α} (hμ : μ ≠ 0)
(hf : ∀ b, measurable (f b)) (hf_int : ∀ b, ∫⁻ a, f b a ∂μ ≠ ∞) (h_directed : directed (≥) f) :
∫⁻ a, ⨅ b, f b a ∂μ = ⨅ b, ∫⁻ a, f b a ∂μ | begin
casesI nonempty_encodable β,
casesI is_empty_or_nonempty β,
{ simp only [with_top.cinfi_empty, lintegral_const],
rw [ennreal.top_mul, if_neg],
simp only [measure.measure_univ_eq_zero, hμ, not_false_iff], },
inhabit β,
have : ∀ a, (⨅ b, f b a) = (⨅ n, f (h_directed.sequence f n) a),
{ refine λ ... | theorem | lintegral_infi_directed_of_measurable | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"countable",
"directed",
"ennreal.top_mul",
"infi_le",
"infi_le_of_le",
"is_empty_or_nonempty",
"le_infi",
"measurable",
"measurable_space",
"nonempty_encodable",
"with_top.cinfi_empty"
] | Monotone convergence for an infimum over a directed family and indexed by a countable type | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_pi_system_Iic [semilattice_inf α] : @is_pi_system α (range Iic) | by { rintros s ⟨us, rfl⟩ t ⟨ut, rfl⟩ _, rw [Iic_inter_Iic], exact ⟨us ⊓ ut, rfl⟩, } | lemma | is_pi_system_Iic | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"is_pi_system",
"semilattice_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_pi_system_Ici [semilattice_sup α] : @is_pi_system α (range Ici) | by { rintros s ⟨us, rfl⟩ t ⟨ut, rfl⟩ _, rw [Ici_inter_Ici], exact ⟨us ⊔ ut, rfl⟩, } | lemma | is_pi_system_Ici | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"is_pi_system",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iic_snd (r : ℝ) : measure α | (ρ.restrict (univ ×ˢ Iic r)).fst | def | measure_theory.measure.Iic_snd | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [] | Measure on `α` such that for a measurable set `s`, `ρ.Iic_snd r s = ρ (s ×ˢ Iic r)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Iic_snd_apply (r : ℝ) {s : set α} (hs : measurable_set s) :
ρ.Iic_snd r s = ρ (s ×ˢ Iic r) | by rw [Iic_snd, fst_apply hs,
restrict_apply' (measurable_set.univ.prod (measurable_set_Iic : measurable_set (Iic r))),
← prod_univ, prod_inter_prod, inter_univ, univ_inter] | lemma | measure_theory.measure.Iic_snd_apply | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"measurable_set",
"measurable_set_Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iic_snd_univ (r : ℝ) : ρ.Iic_snd r univ = ρ (univ ×ˢ Iic r) | Iic_snd_apply ρ r measurable_set.univ | lemma | measure_theory.measure.Iic_snd_univ | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"measurable_set.univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iic_snd_mono {r r' : ℝ} (h_le : r ≤ r') : ρ.Iic_snd r ≤ ρ.Iic_snd r' | begin
intros s hs,
simp_rw Iic_snd_apply ρ _ hs,
refine measure_mono (prod_subset_prod_iff.mpr (or.inl ⟨subset_rfl, Iic_subset_Iic.mpr _⟩)),
exact_mod_cast h_le,
end | lemma | measure_theory.measure.Iic_snd_mono | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iic_snd_le_fst (r : ℝ) : ρ.Iic_snd r ≤ ρ.fst | begin
intros s hs,
simp_rw [fst_apply hs, Iic_snd_apply ρ r hs],
exact measure_mono (prod_subset_preimage_fst _ _),
end | lemma | measure_theory.measure.Iic_snd_le_fst | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iic_snd_ac_fst (r : ℝ) : ρ.Iic_snd r ≪ ρ.fst | measure.absolutely_continuous_of_le (Iic_snd_le_fst ρ r) | lemma | measure_theory.measure.Iic_snd_ac_fst | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_finite_measure.Iic_snd {ρ : measure (α × ℝ)} [is_finite_measure ρ] (r : ℝ) :
is_finite_measure (ρ.Iic_snd r) | is_finite_measure_of_le _ (Iic_snd_le_fst ρ _) | lemma | measure_theory.measure.is_finite_measure.Iic_snd | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi_Iic_snd_gt (t : ℚ) {s : set α} (hs : measurable_set s) [is_finite_measure ρ] :
(⨅ r : {r' : ℚ // t < r'}, ρ.Iic_snd r s) = ρ.Iic_snd t s | begin
simp_rw [ρ.Iic_snd_apply _ hs],
rw ← measure_Inter_eq_infi,
{ rw ← prod_Inter,
congr' with x : 1,
simp only [mem_Inter, mem_Iic, subtype.forall, subtype.coe_mk],
refine ⟨λ h, _, λ h a hta, h.trans _⟩,
{ refine le_of_forall_lt_rat_imp_le (λ q htq, h q _),
exact_mod_cast htq, },
{ ex... | lemma | measure_theory.measure.infi_Iic_snd_gt | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"coe_coe",
"le_of_forall_lt_rat_imp_le",
"lt_add_one",
"measurable_set",
"measurable_set_Iic",
"monotone.directed_ge",
"prod_Inter",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Iic_snd_at_top {s : set α} (hs : measurable_set s) :
tendsto (λ r : ℚ, ρ.Iic_snd r s) at_top (𝓝 (ρ.fst s)) | begin
simp_rw [ρ.Iic_snd_apply _ hs, fst_apply hs, ← prod_univ],
rw [← real.Union_Iic_rat, prod_Union],
refine tendsto_measure_Union (λ r q hr_le_q x, _),
simp only [mem_prod, mem_Iic, and_imp],
refine λ hxs hxr, ⟨hxs, hxr.trans _⟩,
exact_mod_cast hr_le_q,
end | lemma | measure_theory.measure.tendsto_Iic_snd_at_top | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"and_imp",
"measurable_set",
"real.Union_Iic_rat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Iic_snd_at_bot [is_finite_measure ρ] {s : set α} (hs : measurable_set s) :
tendsto (λ r : ℚ, ρ.Iic_snd r s) at_bot (𝓝 0) | begin
simp_rw [ρ.Iic_snd_apply _ hs],
have h_empty : ρ (s ×ˢ ∅) = 0, by simp only [prod_empty, measure_empty],
rw [← h_empty, ← real.Inter_Iic_rat, prod_Inter],
suffices h_neg : tendsto (λ r : ℚ, ρ (s ×ˢ Iic (↑-r))) at_top (𝓝 (ρ (⋂ r : ℚ, s ×ˢ Iic (↑-r)))),
{ have h_inter_eq : (⋂ r : ℚ, s ×ˢ Iic (↑-r)) = (⋂ ... | lemma | measure_theory.measure.tendsto_Iic_snd_at_bot | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"measurable_set",
"measurable_set_Iic",
"prod_Inter",
"rat.cast_eq_id",
"rat.cast_neg",
"real.Inter_Iic_rat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pre_cdf (ρ : measure (α × ℝ)) (r : ℚ) : α → ℝ≥0∞ | measure.rn_deriv (ρ.Iic_snd r) ρ.fst | def | probability_theory.pre_cdf | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [] | `pre_cdf` is the Radon-Nikodym derivative of `ρ.Iic_snd` with respect to `ρ.fst` at each
`r : ℚ`. This function `ℚ → α → ℝ≥0∞` is such that for almost all `a : α`, the function `ℚ → ℝ≥0∞`
satisfies the properties of a cdf (monotone with limit 0 at -∞ and 1 at +∞, right-continuous).
We define this function on `ℚ` and n... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
measurable_pre_cdf {ρ : measure (α × ℝ)} {r : ℚ} : measurable (pre_cdf ρ r) | measure.measurable_rn_deriv _ _ | lemma | probability_theory.measurable_pre_cdf | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_density_pre_cdf (ρ : measure (α × ℝ)) (r : ℚ) [is_finite_measure ρ] :
ρ.fst.with_density (pre_cdf ρ r) = ρ.Iic_snd r | measure.absolutely_continuous_iff_with_density_rn_deriv_eq.mp (measure.Iic_snd_ac_fst ρ r) | lemma | probability_theory.with_density_pre_cdf | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_lintegral_pre_cdf_fst (ρ : measure (α × ℝ)) (r : ℚ) {s : set α}
(hs : measurable_set s) [is_finite_measure ρ] :
∫⁻ x in s, pre_cdf ρ r x ∂ρ.fst = ρ.Iic_snd r s | begin
have : ∀ r, ∫⁻ x in s, pre_cdf ρ r x ∂ρ.fst = ∫⁻ x in s, (pre_cdf ρ r * 1) x ∂ρ.fst,
{ simp only [mul_one, eq_self_iff_true, forall_const], },
rw [this, ← set_lintegral_with_density_eq_set_lintegral_mul _ measurable_pre_cdf _ hs],
{ simp only [with_density_pre_cdf ρ r, pi.one_apply, lintegral_one, measure... | lemma | probability_theory.set_lintegral_pre_cdf_fst | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"forall_const",
"measurable_const",
"measurable_set",
"measurable_set.univ",
"mul_one",
"pi.one_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_pre_cdf (ρ : measure (α × ℝ)) [is_finite_measure ρ] :
∀ᵐ a ∂ρ.fst, monotone (λ r, pre_cdf ρ r a) | begin
simp_rw [monotone, ae_all_iff],
refine λ r r' hrr', ae_le_of_forall_set_lintegral_le_of_sigma_finite
measurable_pre_cdf measurable_pre_cdf (λ s hs hs_fin, _),
rw [set_lintegral_pre_cdf_fst ρ r hs, set_lintegral_pre_cdf_fst ρ r' hs],
refine measure.Iic_snd_mono ρ _ s hs,
exact_mod_cast hrr',
end | lemma | probability_theory.monotone_pre_cdf | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_lintegral_infi_gt_pre_cdf (ρ : measure (α × ℝ)) [is_finite_measure ρ] (t : ℚ)
{s : set α} (hs : measurable_set s) :
∫⁻ x in s, ⨅ r : Ioi t, pre_cdf ρ r x ∂ρ.fst = ρ.Iic_snd t s | begin
refine le_antisymm _ _,
{ have h : ∀ q : Ioi t, ∫⁻ x in s, ⨅ r : Ioi t, pre_cdf ρ r x ∂ρ.fst ≤ ρ.Iic_snd q s,
{ intros q,
rw [coe_coe, ← set_lintegral_pre_cdf_fst ρ _ hs],
refine set_lintegral_mono_ae _ measurable_pre_cdf _,
{ exact measurable_infi (λ _, measurable_pre_cdf), },
{ f... | lemma | probability_theory.set_lintegral_infi_gt_pre_cdf | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"coe_coe",
"infi_le",
"le_infi",
"measurable_infi",
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pre_cdf_le_one (ρ : measure (α × ℝ)) [is_finite_measure ρ] :
∀ᵐ a ∂ρ.fst, ∀ r, pre_cdf ρ r a ≤ 1 | begin
rw ae_all_iff,
refine λ r, ae_le_of_forall_set_lintegral_le_of_sigma_finite measurable_pre_cdf
measurable_const (λ s hs hs_fin, _),
rw set_lintegral_pre_cdf_fst ρ r hs,
simp only [pi.one_apply, lintegral_one, measure.restrict_apply, measurable_set.univ, univ_inter],
exact measure.Iic_snd_le_fst ρ r ... | lemma | probability_theory.pre_cdf_le_one | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"measurable_const",
"measurable_set.univ",
"pi.one_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_lintegral_pre_cdf_at_top (ρ : measure (α × ℝ)) [is_finite_measure ρ] :
tendsto (λ r, ∫⁻ a, pre_cdf ρ r a ∂ρ.fst) at_top (𝓝 (ρ univ)) | begin
convert ρ.tendsto_Iic_snd_at_top measurable_set.univ,
{ ext1 r,
rw [← set_lintegral_univ, set_lintegral_pre_cdf_fst ρ _ measurable_set.univ], },
{ exact measure.fst_univ.symm, },
end | lemma | probability_theory.tendsto_lintegral_pre_cdf_at_top | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"measurable_set.univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_lintegral_pre_cdf_at_bot (ρ : measure (α × ℝ)) [is_finite_measure ρ] :
tendsto (λ r, ∫⁻ a, pre_cdf ρ r a ∂ρ.fst) at_bot (𝓝 0) | begin
convert ρ.tendsto_Iic_snd_at_bot measurable_set.univ,
ext1 r,
rw [← set_lintegral_univ, set_lintegral_pre_cdf_fst ρ _ measurable_set.univ],
end | lemma | probability_theory.tendsto_lintegral_pre_cdf_at_bot | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"measurable_set.univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_pre_cdf_at_top_one (ρ : measure (α × ℝ)) [is_finite_measure ρ] :
∀ᵐ a ∂ρ.fst, tendsto (λ r, pre_cdf ρ r a) at_top (𝓝 1) | begin
-- We show first that `pre_cdf` has a limit almost everywhere. That limit has to be at most 1.
-- We then show that the integral of `pre_cdf` tends to the integral of 1, and that it also tends
-- to the integral of the limit. Since the limit is at most 1 and has same integral as 1, it is
-- equal to 1 a.e... | lemma | probability_theory.tendsto_pre_cdf_at_top_one | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"ae_measurable",
"ae_measurable_of_tendsto_metrizable_ae",
"le_of_tendsto'",
"tendsto_coe_nat_at_top_at_top",
"tendsto_nhds_unique",
"tendsto_of_monotone",
"tsub_eq_zero_iff_le",
"tsub_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_pre_cdf_at_bot_zero (ρ : measure (α × ℝ)) [is_finite_measure ρ] :
∀ᵐ a ∂ρ.fst, tendsto (λ r, pre_cdf ρ r a) at_bot (𝓝 0) | begin
-- We show first that `pre_cdf` has a limit in ℝ≥0∞ almost everywhere.
-- We then show that the integral of `pre_cdf` tends to 0, and that it also tends
-- to the integral of the limit. Since the limit is has integral 0, it is equal to 0 a.e.
suffices : ∀ᵐ a ∂ρ.fst, tendsto (λ r, pre_cdf ρ (-r) a) at_top ... | lemma | probability_theory.tendsto_pre_cdf_at_bot_zero | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"ae_measurable",
"ae_measurable_of_tendsto_metrizable_ae",
"antitone",
"exists_rat_gt",
"measurable_set.univ",
"measurable_set_Iic",
"not_forall",
"prod_Inter",
"tendsto_nhds_unique",
"tendsto_of_antitone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_gt_pre_cdf (ρ : measure (α × ℝ)) [is_finite_measure ρ] :
∀ᵐ a ∂ρ.fst, ∀ t : ℚ, (⨅ r : Ioi t, pre_cdf ρ r a) = pre_cdf ρ t a | begin
rw ae_all_iff,
refine λ t, ae_eq_of_forall_set_lintegral_eq_of_sigma_finite _ measurable_pre_cdf _,
{ exact measurable_infi (λ i, measurable_pre_cdf), },
intros s hs hs_fin,
rw [set_lintegral_infi_gt_pre_cdf ρ t hs, set_lintegral_pre_cdf_fst ρ t hs],
end | lemma | probability_theory.inf_gt_pre_cdf | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"measurable_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_cond_cdf (ρ : measure (α × ℝ)) (a : α) : Prop | (mono : monotone (λ r, pre_cdf ρ r a))
(le_one : ∀ r, pre_cdf ρ r a ≤ 1)
(tendsto_at_top_one : tendsto (λ r, pre_cdf ρ r a) at_top (𝓝 1))
(tendsto_at_bot_zero : tendsto (λ r, pre_cdf ρ r a) at_bot (𝓝 0))
(infi_rat_gt_eq : ∀ t : ℚ, (⨅ r : Ioi t, pre_cdf ρ r a) = pre_cdf ρ t a) | structure | probability_theory.has_cond_cdf | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"monotone"
] | A product measure on `α × ℝ` is said to have a conditional cdf at `a : α` if `pre_cdf` is
monotone with limit 0 at -∞ and 1 at +∞, and is right continuous.
This property holds almost everywhere (see `has_cond_cdf_ae`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_cond_cdf_ae (ρ : measure (α × ℝ)) [is_finite_measure ρ] :
∀ᵐ a ∂ρ.fst, has_cond_cdf ρ a | begin
filter_upwards [monotone_pre_cdf ρ, pre_cdf_le_one ρ, tendsto_pre_cdf_at_top_one ρ,
tendsto_pre_cdf_at_bot_zero ρ, inf_gt_pre_cdf ρ] with a h1 h2 h3 h4 h5,
exact ⟨h1, h2, h3, h4, h5⟩,
end | lemma | probability_theory.has_cond_cdf_ae | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_cdf_set (ρ : measure (α × ℝ)) : set α | (to_measurable ρ.fst {b | ¬ has_cond_cdf ρ b})ᶜ | def | probability_theory.cond_cdf_set | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [] | A measurable set of elements of `α` such that `ρ` has a conditional cdf at all
`a ∈ cond_cdf_set`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
measurable_set_cond_cdf_set (ρ : measure (α × ℝ)) : measurable_set (cond_cdf_set ρ) | (measurable_set_to_measurable _ _).compl | lemma | probability_theory.measurable_set_cond_cdf_set | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_cond_cdf_of_mem_cond_cdf_set {ρ : measure (α × ℝ)} {a : α} (h : a ∈ cond_cdf_set ρ) :
has_cond_cdf ρ a | begin
rw [cond_cdf_set, mem_compl_iff] at h,
have h_ss := subset_to_measurable ρ.fst {b | ¬ has_cond_cdf ρ b},
by_contra ha,
exact h (h_ss ha),
end | lemma | probability_theory.has_cond_cdf_of_mem_cond_cdf_set | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"by_contra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_cond_cdf_set_ae (ρ : measure (α × ℝ)) [is_finite_measure ρ] :
∀ᵐ a ∂ρ.fst, a ∈ cond_cdf_set ρ | begin
simp_rw [ae_iff, cond_cdf_set, not_mem_compl_iff, set_of_mem_eq, measure_to_measurable],
exact has_cond_cdf_ae ρ,
end | lemma | probability_theory.mem_cond_cdf_set_ae | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_cdf_rat (ρ : measure (α × ℝ)) : α → ℚ → ℝ | λ a, if a ∈ cond_cdf_set ρ then (λ r, (pre_cdf ρ r a).to_real) else (λ r, if r < 0 then 0 else 1) | def | probability_theory.cond_cdf_rat | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [] | Conditional cdf of the measure given the value on `α`, restricted to the rationals.
It is defined to be `pre_cdf` if `a ∈ cond_cdf_set`, and a default cdf-like function
otherwise. This is an auxiliary definition used to define `cond_cdf`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cond_cdf_rat_of_not_mem (ρ : measure (α × ℝ)) (a : α) (h : a ∉ cond_cdf_set ρ) {r : ℚ} :
cond_cdf_rat ρ a r = if r < 0 then 0 else 1 | by simp only [cond_cdf_rat, h, if_false] | lemma | probability_theory.cond_cdf_rat_of_not_mem | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_cdf_rat_of_mem (ρ : measure (α × ℝ)) (a : α) (h : a ∈ cond_cdf_set ρ) (r : ℚ) :
cond_cdf_rat ρ a r = (pre_cdf ρ r a).to_real | by simp only [cond_cdf_rat, h, if_true] | lemma | probability_theory.cond_cdf_rat_of_mem | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_cond_cdf_rat (ρ : measure (α × ℝ)) (a : α) :
monotone (cond_cdf_rat ρ a) | begin
by_cases h : a ∈ cond_cdf_set ρ,
{ simp only [cond_cdf_rat, h, if_true, forall_const, and_self],
intros r r' hrr',
have h' := has_cond_cdf_of_mem_cond_cdf_set h,
have h_ne_top : ∀ r, pre_cdf ρ r a ≠ ∞ := λ r, ((h'.le_one r).trans_lt ennreal.one_lt_top).ne,
rw ennreal.to_real_le_to_real (h_ne_t... | lemma | probability_theory.monotone_cond_cdf_rat | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"ennreal.one_lt_top",
"ennreal.to_real_le_to_real",
"forall_const",
"le_rfl",
"monotone",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_cond_cdf_rat (ρ : measure (α × ℝ)) (q : ℚ) :
measurable (λ a, cond_cdf_rat ρ a q) | begin
simp_rw [cond_cdf_rat, ite_apply],
exact measurable.ite (measurable_set_cond_cdf_set ρ) measurable_pre_cdf.ennreal_to_real
measurable_const,
end | lemma | probability_theory.measurable_cond_cdf_rat | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"ite_apply",
"measurable",
"measurable.ite",
"measurable_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_cdf_rat_nonneg (ρ : measure (α × ℝ)) (a : α) (r : ℚ) :
0 ≤ cond_cdf_rat ρ a r | by { unfold cond_cdf_rat, split_ifs, exacts [ennreal.to_real_nonneg, le_rfl, zero_le_one], } | lemma | probability_theory.cond_cdf_rat_nonneg | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"ennreal.to_real_nonneg",
"le_rfl",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_cdf_rat_le_one (ρ : measure (α × ℝ)) (a : α) (r : ℚ) :
cond_cdf_rat ρ a r ≤ 1 | begin
unfold cond_cdf_rat,
split_ifs with h,
{ refine ennreal.to_real_le_of_le_of_real zero_le_one _,
rw ennreal.of_real_one,
exact (has_cond_cdf_of_mem_cond_cdf_set h).le_one r, },
exacts [zero_le_one, le_rfl],
end | lemma | probability_theory.cond_cdf_rat_le_one | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"ennreal.of_real_one",
"ennreal.to_real_le_of_le_of_real",
"le_rfl",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_cond_cdf_rat_at_bot (ρ : measure (α × ℝ)) (a : α) :
tendsto (cond_cdf_rat ρ a) at_bot (𝓝 0) | begin
unfold cond_cdf_rat,
split_ifs with h,
{ rw [← ennreal.zero_to_real, ennreal.tendsto_to_real_iff],
{ exact (has_cond_cdf_of_mem_cond_cdf_set h).tendsto_at_bot_zero, },
{ have h' := has_cond_cdf_of_mem_cond_cdf_set h,
exact λ r, ((h'.le_one r).trans_lt ennreal.one_lt_top).ne, },
{ exact enn... | lemma | probability_theory.tendsto_cond_cdf_rat_at_bot | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"ennreal.one_lt_top",
"ennreal.tendsto_to_real_iff",
"ennreal.zero_ne_top",
"ennreal.zero_to_real",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_cond_cdf_rat_at_top (ρ : measure (α × ℝ)) (a : α) :
tendsto (cond_cdf_rat ρ a) at_top (𝓝 1) | begin
unfold cond_cdf_rat,
split_ifs with h,
{ have h' := has_cond_cdf_of_mem_cond_cdf_set h,
rw [← ennreal.one_to_real, ennreal.tendsto_to_real_iff],
{ exact h'.tendsto_at_top_one, },
{ exact λ r, ((h'.le_one r).trans_lt ennreal.one_lt_top).ne, },
{ exact ennreal.one_ne_top, }, },
{ refine (ten... | lemma | probability_theory.tendsto_cond_cdf_rat_at_top | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"ennreal.one_lt_top",
"ennreal.one_ne_top",
"ennreal.one_to_real",
"ennreal.tendsto_to_real_iff",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_cdf_rat_ae_eq (ρ : measure (α × ℝ)) [is_finite_measure ρ] (r : ℚ) :
(λ a, cond_cdf_rat ρ a r) =ᵐ[ρ.fst] λ a, (pre_cdf ρ r a).to_real | by filter_upwards [mem_cond_cdf_set_ae ρ] with a ha using cond_cdf_rat_of_mem ρ a ha r | lemma | probability_theory.cond_cdf_rat_ae_eq | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_real_cond_cdf_rat_ae_eq (ρ : measure (α × ℝ)) [is_finite_measure ρ] (r : ℚ) :
(λ a, ennreal.of_real (cond_cdf_rat ρ a r)) =ᵐ[ρ.fst] pre_cdf ρ r | begin
filter_upwards [cond_cdf_rat_ae_eq ρ r, pre_cdf_le_one ρ] with a ha ha_le_one,
rw [ha, ennreal.of_real_to_real],
exact ((ha_le_one r).trans_lt ennreal.one_lt_top).ne,
end | lemma | probability_theory.of_real_cond_cdf_rat_ae_eq | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"ennreal.of_real",
"ennreal.of_real_to_real",
"ennreal.one_lt_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_gt_cond_cdf_rat (ρ : measure (α × ℝ)) (a : α) (t : ℚ) :
(⨅ r : Ioi t, cond_cdf_rat ρ a r) = cond_cdf_rat ρ a t | begin
by_cases ha : a ∈ cond_cdf_set ρ,
{ simp_rw cond_cdf_rat_of_mem ρ a ha,
have ha' := has_cond_cdf_of_mem_cond_cdf_set ha,
rw ← ennreal.to_real_infi,
{ suffices : (⨅ (i : ↥(Ioi t)), pre_cdf ρ ↑i a) = pre_cdf ρ t a, by rw this,
rw ← ha'.infi_rat_gt_eq, },
{ exact λ r, ((ha'.le_one r).trans_... | lemma | probability_theory.inf_gt_cond_cdf_rat | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"bdd_below",
"cinfi_le",
"ennreal.one_lt_top",
"ennreal.to_real_infi",
"le_cinfi",
"le_rfl",
"lt_add_one",
"subtype.coe_mk",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_cdf' (ρ : measure (α × ℝ)) : α → ℝ → ℝ | λ a t, ⨅ r : {r' : ℚ // t < r'}, cond_cdf_rat ρ a r | def | probability_theory.cond_cdf' | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [] | Conditional cdf of the measure given the value on `α`, as a plain function. This is an auxiliary
definition used to define `cond_cdf`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cond_cdf'_def {ρ : measure (α × ℝ)} {a : α} {x : ℝ} :
cond_cdf' ρ a x = ⨅ r : {r : ℚ // x < r}, cond_cdf_rat ρ a r | by rw cond_cdf' | lemma | probability_theory.cond_cdf'_def | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_cdf'_eq_cond_cdf_rat (ρ : measure (α × ℝ)) (a : α) (r : ℚ) :
cond_cdf' ρ a r = cond_cdf_rat ρ a r | begin
rw [← inf_gt_cond_cdf_rat ρ a r, cond_cdf'],
refine equiv.infi_congr _ _,
{ exact
{ to_fun := λ t, ⟨t.1, by exact_mod_cast t.2⟩,
inv_fun := λ t, ⟨t.1, by exact_mod_cast t.2⟩,
left_inv := λ t, by simp only [subtype.val_eq_coe, subtype.coe_eta],
right_inv := λ t, by simp only [subtype.va... | lemma | probability_theory.cond_cdf'_eq_cond_cdf_rat | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"equiv.coe_fn_mk",
"equiv.infi_congr",
"inv_fun",
"subtype.coe_eta",
"subtype.coe_mk",
"subtype.val_eq_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_cdf'_nonneg (ρ : measure (α × ℝ)) (a : α) (r : ℝ) :
0 ≤ cond_cdf' ρ a r | begin
haveI : nonempty {r' : ℚ // r < ↑r'},
{ obtain ⟨r, hrx⟩ := exists_rat_gt r,
exact ⟨⟨r, hrx⟩⟩, },
rw cond_cdf'_def,
exact le_cinfi (λ r', cond_cdf_rat_nonneg ρ a _),
end | lemma | probability_theory.cond_cdf'_nonneg | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"exists_rat_gt",
"le_cinfi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_below_range_cond_cdf_rat_gt (ρ : measure (α × ℝ)) (a : α) (x : ℝ) :
bdd_below (range (λ (r : {r' : ℚ // x < ↑r'}), cond_cdf_rat ρ a r)) | by { refine ⟨0, λ z, _⟩, rintros ⟨u, rfl⟩, exact cond_cdf_rat_nonneg ρ a _, } | lemma | probability_theory.bdd_below_range_cond_cdf_rat_gt | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"bdd_below"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_cond_cdf' (ρ : measure (α × ℝ)) (a : α) : monotone (cond_cdf' ρ a) | begin
intros x y hxy,
haveI : nonempty {r' : ℚ // y < ↑r'},
{ obtain ⟨r, hrx⟩ := exists_rat_gt y,
exact ⟨⟨r, hrx⟩⟩, },
simp_rw cond_cdf'_def,
refine le_cinfi (λ r, (cinfi_le _ _).trans_eq _),
{ exact ⟨r.1, hxy.trans_lt r.prop⟩, },
{ exact bdd_below_range_cond_cdf_rat_gt ρ a x, },
{ refl, },
end | lemma | probability_theory.monotone_cond_cdf' | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"cinfi_le",
"exists_rat_gt",
"le_cinfi",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_cond_cdf'_Ici (ρ : measure (α × ℝ)) (a : α) (x : ℝ) :
continuous_within_at (cond_cdf' ρ a) (Ici x) x | begin
rw ← continuous_within_at_Ioi_iff_Ici,
convert monotone.tendsto_nhds_within_Ioi (monotone_cond_cdf' ρ a) x,
rw Inf_image',
have h' : (⨅ r : Ioi x, cond_cdf' ρ a r) = ⨅ r : {r' : ℚ // x < r'}, cond_cdf' ρ a r,
{ refine infi_Ioi_eq_infi_rat_gt x _ (monotone_cond_cdf' ρ a),
refine ⟨0, λ z, _⟩,
rint... | lemma | probability_theory.continuous_within_at_cond_cdf'_Ici | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"Inf_image'",
"continuous_within_at",
"continuous_within_at_Ioi_iff_Ici",
"infi_Ioi_eq_infi_rat_gt",
"monotone.tendsto_nhds_within_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_cdf (ρ : measure (α × ℝ)) (a : α) : stieltjes_function | { to_fun := cond_cdf' ρ a,
mono' := monotone_cond_cdf' ρ a,
right_continuous' := λ x, continuous_within_at_cond_cdf'_Ici ρ a x, } | def | probability_theory.cond_cdf | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"stieltjes_function"
] | Conditional cdf of the measure given the value on `α`, as a Stieltjes function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cond_cdf_eq_cond_cdf_rat (ρ : measure (α × ℝ)) (a : α) (r : ℚ) :
cond_cdf ρ a r = cond_cdf_rat ρ a r | cond_cdf'_eq_cond_cdf_rat ρ a r | lemma | probability_theory.cond_cdf_eq_cond_cdf_rat | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_cdf_nonneg (ρ : measure (α × ℝ)) (a : α) (r : ℝ) :
0 ≤ cond_cdf ρ a r | cond_cdf'_nonneg ρ a r | lemma | probability_theory.cond_cdf_nonneg | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [] | The conditional cdf is non-negative for all `a : α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cond_cdf_le_one (ρ : measure (α × ℝ)) (a : α) (x : ℝ) :
cond_cdf ρ a x ≤ 1 | begin
obtain ⟨r, hrx⟩ := exists_rat_gt x,
rw ← stieltjes_function.infi_rat_gt_eq,
simp_rw [coe_coe, cond_cdf_eq_cond_cdf_rat],
refine cinfi_le_of_le (bdd_below_range_cond_cdf_rat_gt ρ a x) _ (cond_cdf_rat_le_one _ _ _),
exact ⟨r, hrx⟩,
end | lemma | probability_theory.cond_cdf_le_one | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"cinfi_le_of_le",
"coe_coe",
"exists_rat_gt",
"stieltjes_function.infi_rat_gt_eq"
] | The conditional cdf is lower or equal to 1 for all `a : α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_cond_cdf_at_bot (ρ : measure (α × ℝ)) (a : α) :
tendsto (cond_cdf ρ a) at_bot (𝓝 0) | begin
have h_exists : ∀ x : ℝ, ∃ q : ℚ, x < q ∧ ↑q < x + 1 := λ x, exists_rat_btwn (lt_add_one x),
let qs : ℝ → ℚ := λ x, (h_exists x).some,
have hqs_tendsto : tendsto qs at_bot at_bot,
{ rw tendsto_at_bot_at_bot,
refine λ q, ⟨q - 1, λ y hy, _⟩,
have h_le : ↑(qs y) ≤ (q : ℝ) - 1 + 1 :=
((h_exists ... | lemma | probability_theory.tendsto_cond_cdf_at_bot | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"exists_rat_btwn",
"function.comp_apply",
"le_rfl",
"lt_add_one",
"tendsto_const_nhds",
"tendsto_of_tendsto_of_tendsto_of_le_of_le"
] | The conditional cdf tends to 0 at -∞ for all `a : α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_cond_cdf_at_top (ρ : measure (α × ℝ)) (a : α) :
tendsto (cond_cdf ρ a) at_top (𝓝 1) | begin
have h_exists : ∀ x : ℝ, ∃ q : ℚ, x-1 < q ∧ ↑q < x := λ x, exists_rat_btwn (sub_one_lt x),
let qs : ℝ → ℚ := λ x, (h_exists x).some,
have hqs_tendsto : tendsto qs at_top at_top,
{ rw tendsto_at_top_at_top,
refine λ q, ⟨q + 1, λ y hy, _⟩,
have h_le : y - 1 ≤ qs y := (h_exists y).some_spec.1.le,
... | lemma | probability_theory.tendsto_cond_cdf_at_top | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"exists_rat_btwn",
"function.comp_apply",
"sub_one_lt",
"tendsto_const_nhds",
"tendsto_of_tendsto_of_tendsto_of_le_of_le"
] | The conditional cdf tends to 1 at +∞ for all `a : α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cond_cdf_ae_eq (ρ : measure (α × ℝ)) [is_finite_measure ρ] (r : ℚ) :
(λ a, cond_cdf ρ a r) =ᵐ[ρ.fst] λ a, (pre_cdf ρ r a).to_real | by filter_upwards [mem_cond_cdf_set_ae ρ] with a ha
using (cond_cdf_eq_cond_cdf_rat ρ a r).trans (cond_cdf_rat_of_mem ρ a ha r) | lemma | probability_theory.cond_cdf_ae_eq | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_real_cond_cdf_ae_eq (ρ : measure (α × ℝ)) [is_finite_measure ρ] (r : ℚ) :
(λ a, ennreal.of_real (cond_cdf ρ a r)) =ᵐ[ρ.fst] pre_cdf ρ r | begin
filter_upwards [cond_cdf_ae_eq ρ r, pre_cdf_le_one ρ] with a ha ha_le_one,
rw [ha, ennreal.of_real_to_real],
exact ((ha_le_one r).trans_lt ennreal.one_lt_top).ne,
end | lemma | probability_theory.of_real_cond_cdf_ae_eq | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"ennreal.of_real",
"ennreal.of_real_to_real",
"ennreal.one_lt_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_cond_cdf (ρ : measure (α × ℝ)) (x : ℝ) :
measurable (λ a, cond_cdf ρ a x) | begin
have : (λ a, cond_cdf ρ a x) = λ a, (⨅ (r : {r' // x < ↑r'}), cond_cdf_rat ρ a ↑r),
{ ext1 a,
rw ← stieltjes_function.infi_rat_gt_eq,
congr' with q,
rw [coe_coe, cond_cdf_eq_cond_cdf_rat], },
rw this,
exact measurable_cinfi (λ q, measurable_cond_cdf_rat ρ q)
(λ a, bdd_below_range_cond_cdf_... | lemma | probability_theory.measurable_cond_cdf | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"coe_coe",
"measurable",
"measurable_cinfi",
"stieltjes_function.infi_rat_gt_eq"
] | The conditional cdf is a measurable function of `a : α` for all `x : ℝ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_lintegral_cond_cdf_rat (ρ : measure (α × ℝ)) [is_finite_measure ρ] (r : ℚ)
{s : set α} (hs : measurable_set s) :
∫⁻ a in s, ennreal.of_real (cond_cdf ρ a r) ∂ρ.fst = ρ (s ×ˢ Iic r) | begin
have : ∀ᵐ a ∂ρ.fst, a ∈ s → ennreal.of_real (cond_cdf ρ a r) = pre_cdf ρ r a,
{ filter_upwards [of_real_cond_cdf_ae_eq ρ r] with a ha using λ _, ha, },
rw [set_lintegral_congr_fun hs this, set_lintegral_pre_cdf_fst ρ r hs],
exact ρ.Iic_snd_apply r hs,
end | lemma | probability_theory.set_lintegral_cond_cdf_rat | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"ennreal.of_real",
"measurable_set"
] | Auxiliary lemma for `set_lintegral_cond_cdf`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_lintegral_cond_cdf (ρ : measure (α × ℝ)) [is_finite_measure ρ] (x : ℝ)
{s : set α} (hs : measurable_set s) :
∫⁻ a in s, ennreal.of_real (cond_cdf ρ a x) ∂ρ.fst = ρ (s ×ˢ Iic x) | begin
-- We have the result for `x : ℚ` thanks to `set_lintegral_cond_cdf_rat`. We use the equality
-- `cond_cdf ρ a x = ⨅ r : {r' : ℚ // x < r'}, cond_cdf ρ a r` and a monotone convergence
-- argument to extend it to the reals.
by_cases hρ_zero : ρ.fst.restrict s = 0,
{ rw [hρ_zero, lintegral_zero_measure],
... | lemma | probability_theory.set_lintegral_cond_cdf | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"ennreal.of_real",
"ennreal.of_real_cinfi",
"ennreal.of_real_le_of_real",
"exists_rat_gt",
"lintegral_infi_directed_of_measurable",
"measurable_set",
"measurable_set_Iic",
"monotone.directed_ge",
"prod_Inter",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lintegral_cond_cdf (ρ : measure (α × ℝ)) [is_finite_measure ρ] (x : ℝ) :
∫⁻ a, ennreal.of_real (cond_cdf ρ a x) ∂ρ.fst = ρ (univ ×ˢ Iic x) | by rw [← set_lintegral_univ, set_lintegral_cond_cdf ρ _ measurable_set.univ] | lemma | probability_theory.lintegral_cond_cdf | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"ennreal.of_real",
"measurable_set.univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strongly_measurable_cond_cdf (ρ : measure (α × ℝ)) (x : ℝ) :
strongly_measurable (λ a, cond_cdf ρ a x) | (measurable_cond_cdf ρ x).strongly_measurable | lemma | probability_theory.strongly_measurable_cond_cdf | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [] | The conditional cdf is a strongly measurable function of `a : α` for all `x : ℝ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integrable_cond_cdf (ρ : measure (α × ℝ)) [is_finite_measure ρ] (x : ℝ) :
integrable (λ a, cond_cdf ρ a x) ρ.fst | begin
refine integrable_of_forall_fin_meas_le _ (measure_lt_top ρ.fst univ) _ (λ t ht hρt, _),
{ exact (strongly_measurable_cond_cdf ρ _).ae_strongly_measurable, },
{ have : ∀ y, (‖cond_cdf ρ y x‖₊ : ℝ≥0∞) ≤ 1,
{ intro y,
rw real.nnnorm_of_nonneg (cond_cdf_nonneg _ _ _),
exact_mod_cast cond_cdf_le... | lemma | probability_theory.integrable_cond_cdf | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"measurable_one",
"measurable_set.univ",
"pi.one_apply",
"real.nnnorm_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_integral_cond_cdf (ρ : measure (α × ℝ)) [is_finite_measure ρ] (x : ℝ)
{s : set α} (hs : measurable_set s) :
∫ a in s, cond_cdf ρ a x ∂ρ.fst = (ρ (s ×ˢ Iic x)).to_real | begin
have h := set_lintegral_cond_cdf ρ x hs,
rw ← of_real_integral_eq_lintegral_of_real at h,
{ rw [← h, ennreal.to_real_of_real],
exact integral_nonneg (λ _, cond_cdf_nonneg _ _ _), },
{ exact (integrable_cond_cdf _ _).integrable_on, },
{ exact eventually_of_forall (λ _, cond_cdf_nonneg _ _ _), },
end | lemma | probability_theory.set_integral_cond_cdf | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"ennreal.to_real_of_real",
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_cond_cdf (ρ : measure (α × ℝ)) [is_finite_measure ρ] (x : ℝ) :
∫ a, cond_cdf ρ a x ∂ρ.fst = (ρ (univ ×ˢ Iic x)).to_real | by rw [← set_integral_cond_cdf ρ _ measurable_set.univ, measure.restrict_univ] | lemma | probability_theory.integral_cond_cdf | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"measurable_set.univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measure_cond_cdf_Iic (ρ : measure (α × ℝ)) (a : α) (x : ℝ) :
(cond_cdf ρ a).measure (Iic x) = ennreal.of_real (cond_cdf ρ a x) | begin
rw [← sub_zero (cond_cdf ρ a x)],
exact (cond_cdf ρ a).measure_Iic (tendsto_cond_cdf_at_bot ρ a) _,
end | lemma | probability_theory.measure_cond_cdf_Iic | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"ennreal.of_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measure_cond_cdf_univ (ρ : measure (α × ℝ)) (a : α) :
(cond_cdf ρ a).measure univ = 1 | begin
rw [← ennreal.of_real_one, ← sub_zero (1 : ℝ)],
exact stieltjes_function.measure_univ _ (tendsto_cond_cdf_at_bot ρ a)
(tendsto_cond_cdf_at_top ρ a),
end | lemma | probability_theory.measure_cond_cdf_univ | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"ennreal.of_real_one",
"stieltjes_function.measure_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_measure_cond_cdf (ρ : measure (α × ℝ)) :
measurable (λ a, (cond_cdf ρ a).measure) | begin
rw measure.measurable_measure,
refine λ s hs, measurable_space.induction_on_inter
(borel_eq_generate_from_Iic ℝ) is_pi_system_Iic _ _ _ _ hs,
{ simp only [measure_empty, measurable_const], },
{ rintros S ⟨u, rfl⟩,
simp_rw measure_cond_cdf_Iic ρ _ u,
exact (measurable_cond_cdf ρ u).ennreal_of_r... | lemma | probability_theory.measurable_measure_cond_cdf | probability.kernel | src/probability/kernel/cond_cdf.lean | [
"measure_theory.measure.stieltjes",
"probability.kernel.composition",
"measure_theory.decomposition.radon_nikodym"
] | [
"borel_eq_generate_from_Iic",
"is_pi_system_Iic",
"measurable",
"measurable.ennreal_tsum",
"measurable_const",
"measurable_space.induction_on_inter"
] | The function `a ↦ (cond_cdf ρ a).measure` is measurable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cond_distrib {mα : measurable_space α} [measurable_space β]
(Y : α → Ω) (X : α → β) (μ : measure α) [is_finite_measure μ] :
kernel β Ω | (μ.map (λ a, (X a, Y a))).cond_kernel | def | probability_theory.cond_distrib | probability.kernel | src/probability/kernel/cond_distrib.lean | [
"probability.kernel.disintegration",
"probability.notation"
] | [
"measurable_space"
] | **Regular conditional probability distribution**: kernel associated with the conditional
expectation of `Y` given `X`.
For almost all `a`, `cond_distrib Y X μ` evaluated at `X a` and a measurable set `s` is equal to
the conditional expectation `μ⟦Y ⁻¹' s | mβ.comap X⟧ a`. It also satisfies the equality
`μ[(λ a, f (X a,... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
measurable_cond_distrib (hs : measurable_set s) :
measurable[mβ.comap X] (λ a, cond_distrib Y X μ (X a) s) | (kernel.measurable_coe _ hs).comp (measurable.of_comap_le le_rfl) | lemma | probability_theory.measurable_cond_distrib | probability.kernel | src/probability/kernel/cond_distrib.lean | [
"probability.kernel.disintegration",
"probability.notation"
] | [
"le_rfl",
"measurable",
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.measure_theory.ae_strongly_measurable.ae_integrable_cond_distrib_map_iff
(hX : ae_measurable X μ) (hY : ae_measurable Y μ)
(hf : ae_strongly_measurable f (μ.map (λ a, (X a, Y a)))) :
(∀ᵐ a ∂(μ.map X), integrable (λ ω, f (a, ω)) (cond_distrib Y X μ a))
∧ integrable (λ a, ∫ ω, ‖f (a, ω)‖ ∂(cond_distrib Y... | by rw [cond_distrib, ← hf.ae_integrable_cond_kernel_iff, measure.fst_map_prod_mk₀ hX hY] | lemma | measure_theory.ae_strongly_measurable.ae_integrable_cond_distrib_map_iff | probability.kernel | src/probability/kernel/cond_distrib.lean | [
"probability.kernel.disintegration",
"probability.notation"
] | [
"ae_measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.measure_theory.ae_strongly_measurable.integral_cond_distrib_map
(hX : ae_measurable X μ) (hY : ae_measurable Y μ)
(hf : ae_strongly_measurable f (μ.map (λ a, (X a, Y a)))) :
ae_strongly_measurable (λ x, ∫ y, f (x, y) ∂(cond_distrib Y X μ x)) (μ.map X) | by { rw [← measure.fst_map_prod_mk₀ hX hY, cond_distrib], exact hf.integral_cond_kernel, } | lemma | measure_theory.ae_strongly_measurable.integral_cond_distrib_map | probability.kernel | src/probability/kernel/cond_distrib.lean | [
"probability.kernel.disintegration",
"probability.notation"
] | [
"ae_measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.measure_theory.ae_strongly_measurable.integral_cond_distrib
(hX : ae_measurable X μ) (hY : ae_measurable Y μ)
(hf : ae_strongly_measurable f (μ.map (λ a, (X a, Y a)))) :
ae_strongly_measurable (λ a, ∫ y, f (X a, y) ∂(cond_distrib Y X μ (X a))) μ | (hf.integral_cond_distrib_map hX hY).comp_ae_measurable hX | lemma | measure_theory.ae_strongly_measurable.integral_cond_distrib | probability.kernel | src/probability/kernel/cond_distrib.lean | [
"probability.kernel.disintegration",
"probability.notation"
] | [
"ae_measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ae_strongly_measurable'_integral_cond_distrib
(hX : ae_measurable X μ) (hY : ae_measurable Y μ)
(hf : ae_strongly_measurable f (μ.map (λ a, (X a, Y a)))) :
ae_strongly_measurable' (mβ.comap X) (λ a, ∫ y, f (X a, y) ∂(cond_distrib Y X μ (X a))) μ | (hf.integral_cond_distrib_map hX hY).comp_ae_measurable' hX | lemma | probability_theory.ae_strongly_measurable'_integral_cond_distrib | probability.kernel | src/probability/kernel/cond_distrib.lean | [
"probability.kernel.disintegration",
"probability.notation"
] | [
"ae_measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_to_real_cond_distrib (hX : ae_measurable X μ) (hs : measurable_set s) :
integrable (λ a, (cond_distrib Y X μ (X a) s).to_real) μ | begin
refine integrable_to_real_of_lintegral_ne_top _ _,
{ exact measurable.comp_ae_measurable (kernel.measurable_coe _ hs) hX, },
{ refine ne_of_lt _,
calc ∫⁻ a, cond_distrib Y X μ (X a) s ∂μ
≤ ∫⁻ a, 1 ∂μ : lintegral_mono (λ a, prob_le_one)
... = μ univ : lintegral_one
... < ∞ : measure_lt_to... | lemma | probability_theory.integrable_to_real_cond_distrib | probability.kernel | src/probability/kernel/cond_distrib.lean | [
"probability.kernel.disintegration",
"probability.notation"
] | [
"ae_measurable",
"measurable.comp_ae_measurable",
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.measure_theory.integrable.cond_distrib_ae_map
(hX : ae_measurable X μ) (hY : ae_measurable Y μ)
(hf_int : integrable f (μ.map (λ a, (X a, Y a)))) :
∀ᵐ b ∂(μ.map X), integrable (λ ω, f (b, ω)) (cond_distrib Y X μ b) | by { rw [cond_distrib, ← measure.fst_map_prod_mk₀ hX hY], exact hf_int.cond_kernel_ae, } | lemma | measure_theory.integrable.cond_distrib_ae_map | probability.kernel | src/probability/kernel/cond_distrib.lean | [
"probability.kernel.disintegration",
"probability.notation"
] | [
"ae_measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.measure_theory.integrable.cond_distrib_ae
(hX : ae_measurable X μ) (hY : ae_measurable Y μ)
(hf_int : integrable f (μ.map (λ a, (X a, Y a)))) :
∀ᵐ a ∂μ, integrable (λ ω, f (X a, ω)) (cond_distrib Y X μ (X a)) | ae_of_ae_map hX (hf_int.cond_distrib_ae_map hX hY) | lemma | measure_theory.integrable.cond_distrib_ae | probability.kernel | src/probability/kernel/cond_distrib.lean | [
"probability.kernel.disintegration",
"probability.notation"
] | [
"ae_measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.measure_theory.integrable.integral_norm_cond_distrib_map
(hX : ae_measurable X μ) (hY : ae_measurable Y μ)
(hf_int : integrable f (μ.map (λ a, (X a, Y a)))) :
integrable (λ x, ∫ y, ‖f (x, y)‖ ∂(cond_distrib Y X μ x)) (μ.map X) | by { rw [cond_distrib, ← measure.fst_map_prod_mk₀ hX hY], exact hf_int.integral_norm_cond_kernel, } | lemma | measure_theory.integrable.integral_norm_cond_distrib_map | probability.kernel | src/probability/kernel/cond_distrib.lean | [
"probability.kernel.disintegration",
"probability.notation"
] | [
"ae_measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.measure_theory.integrable.integral_norm_cond_distrib
(hX : ae_measurable X μ) (hY : ae_measurable Y μ)
(hf_int : integrable f (μ.map (λ a, (X a, Y a)))) :
integrable (λ a, ∫ y, ‖f (X a, y)‖ ∂(cond_distrib Y X μ (X a))) μ | (hf_int.integral_norm_cond_distrib_map hX hY).comp_ae_measurable hX | lemma | measure_theory.integrable.integral_norm_cond_distrib | probability.kernel | src/probability/kernel/cond_distrib.lean | [
"probability.kernel.disintegration",
"probability.notation"
] | [
"ae_measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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